Evaluate log base 5 of square root of 8

Problem

(log_5)(√(,8))

Solution

1. Rewrite the square root as an exponent using the property √(,x)=x(1/2)

(log_5)(8(1/2))

2. Apply the power rule for logarithms, (log_b)(xp)=p*(log_b)(x) to move the exponent to the front.

1/2*(log_5)(8)

3. Rewrite the number 8 as a power of 2, since 8=2

1/2*(log_5)(2)

4. Apply the power rule again to move the exponent 3 to the front and multiply it by the existing coefficient.

3/2*(log_5)(2)

5. Apply the change of base formula if a decimal approximation or a different base is required, using (log_b)(a)=ln(a)/ln(b)

(3*ln(2))/(2*ln(5))

Final Answer

(log_5)(√(,8))=3/2*(log_5)(2)

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